Let $\pi:X \to \mathbb{P}^1$ be a regular elliptic surface with a section $\sigma$.
I am interested in this claim:
$\mathbb{P}H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(n))\cong \operatorname{Sym}^n(\sigma) $ by associating to a section the set of points where it vanishes.
The statement only makes sense I guess if $n\ge 0$ (otherwise the left hand side would be empty). I feel like it is true in a more general setting (outside the scope of elliptic surfaces) but don't know where to start.
Thank you for any hints or references.